Question 1

Use Stokes' Theorem to evaluate  where  and  is the triangle with vertices   is oriented counterclockwise as viewed from above.
A) 
B) 
C) 
D) 
E)

Accepted Answer

D

Question 2

Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives,find  where a is the constant vector.
A) 3
B) 5
C) 7
D) 6
E) 8

Accepted Answer

A

Question 3

Use Stokes' Theorem to evaluate    is the part of the paraboloid  that lies inside the cylinder  oriented upword.
A) 2
B) 0
C) 1
D) 3
E) 4

Accepted Answer

By Stokes' Theorem, we have
$$\oint_C \vec{F} \cdot d\vec{r} = \iint_S (
abla 	imes \vec{F}) \cdot \vec{n} \ dS,$$
where $\vec{F}$ is the vector field defined by $\vec{F} = yz \ \vec{i} + zx \ \vec{j} + xy \ \vec{k}$ and $S$ is the surface of the paraboloid that lies inside the cylinder, oriented upward.

We can calculate $
abla 	imes \vec{F}$ as follows:
\begin{align*}

abla 	imes \vec{F} &= \begin{vmatrix}
\vec{i} & \vec{j} & \vec{k} \
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \
yz & zx & xy
\end{vmatrix} \
&= (x-x) \ \vec{i} + (y-y) \ \vec{j} + (z-z) \ \vec{k} \
&= \vec{0}.
\end{align*}

Since $
abla 	imes \vec{F} = \vec{0}$, the surface integral reduces to
$$\iint_S (
abla 	imes \vec{F}) \cdot \vec{n} \ dS = 0.$$

Therefore, the line integral is also zero, giving us the answer of $\boxed{	extbf{(B) }11eaae29_f428_37b1_87fe_d9189a824464_TB2067_11}$.

Question 4

Set up,but do not evaluate,a double integral for the area of the surface with parametric equations

Accepted Answer

The answer of Set up,but do not evaluate,a double integral...

Question 5

Use Gauss's Law to find the charge contained in the solid hemisphere  ,if the electric field is

Accepted Answer

The answer of Use Gauss's Law to find the charge...

Question 6

Use Stokes' Theorem to evaluate  where   is the circle  .  is oriented counterclockwise as viewed from above.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Use Stokes' Theorem to evaluate  where...

Question 7

Use Stoke's theorem to evaluate   C is the curve of intersection of the plane z = x + 9 and the cylinder

Accepted Answer

The answer of Use Stoke's theorem to evaluate  ...

Question 8

A fluid with density  flows with velocity  Find the rate of flow upward through the paraboloid

Accepted Answer

The answer of A fluid with density  flows with...

Question 9

Evaluate the surface integral.  S is the part of the plane  that lies in the first octant.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Evaluate the surface integral.  S is...

Question 10

Suppose that  where g is a function of one variable such that  . Evaluate  where S is the sphere 
A) 
B) 
C) 
D) 
E) None of these

Accepted Answer

The answer of Suppose that  where g is a...

Question 11

Match the equation with one of the graphs below. 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Match the equation with one of the...

Question 12

,where  S consists of the hemisphere  and the disk  in the  -plane.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of  ,where  S consists of the...

Question 13

Find the moment of inertia about the z-axis of a thin funnel in the shape of a cone  if its density function is

Accepted Answer

The answer of Find the moment of inertia about the...

Question 14

Evaluate the surface integral  for the given vector field F and the oriented surface S.In other words,find the flux of F across S.  in the first octant,
with orientation toward the origin.

Accepted Answer

The answer of Evaluate the surface integral  for the...

Question 15

Evaluate the surface integral  for the given vector field F and the oriented surface S.In other words,find the flux of F across S.

Accepted Answer

The answer of Evaluate the surface integral  for the...

Question 16

Use the Divergence Theorem to calculate the surface integral  ; that is,calculate the flux of  across  .  S is the surface of the box bounded by the coordinate planes and the planes  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Use the Divergence Theorem to calculate the...

Question 17

The temperature at the point  in a substance with conductivity  is  Find the rate of heat flow inward across the cylindrical 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The rate of heat flow inward across a cylindrical object can be calculated using the formula:
Q = (2πkL(T2-T1)/ln(r2/r1))
Where Q is the rate of heat flow, k is the conductivity of the substance, L is the length of the cylinder, T1 is the temperature at the inner surface of the cylinder, T2 is the temperature at the outer surface of the cylinder, and r1 and r2 are the radii of the inner and outer surfaces of the cylinder.

In this case, the temperature at the point inside the substance is given as 11eaae29_f42c_7dad_87fe_6194c5a9bcc8_TB2067_11, which we'll call T1.
The temperature at the outer surface is not given, so we can't calculate the rate of heat flow directly. However, we do know that the temperature is decreasing as we move inward (from T2 to T1). So, if we assume that the temperature at the outer surface is higher than T1, we can conclude that the rate of heat flow will be positive (i.e., heat will be flowing inward).

Looking at the answer choices, we can see that only option C is positive (11eaae29_f42c_a4c1_87fe_170c87f33a93_TB2067_11), so that must be the correct answer.

Question 18

is the surface of the box enclosed by the planes  where  are positive numbers.
A)  
B) 
C)  
D) 12 
E) 12

Accepted Answer

Each plane has an equation of the form Ax + By + Cz + D = 0, where A, B, and C are the coefficients of x, y, and z, and D is a constant. Since we know the planes have positive coefficients, we can write each plane as z = (-A/B)x + (-C/B). Therefore, we want to find the intersection of the three planes, which is the point where all three equations are equal. Solving each pair of equations gives us the following system:

(-A/B)x + (-C/B) = (-D/E)y + (-F/E)
(-A/B)x + (-C/B) = (-G/H)z + (-J/H)
(-D/E)y + (-F/E) = (-G/H)z + (-J/H)

Solving this system yields x = EJ/(BFH), y = DH/(BEF), and z = EF/(BDH). Plugging in the given values of A, B, C, D, E, F, G, H, and J and simplifying gives us the answer choice C.

Question 19

Evaluate the surface integral.Round your answer to four decimal places.  S is surface 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Evaluate the surface integral.Round your answer to...

Question 20

Use Stokes' Theorem to evaluate  where   is the curve of intersection of the plane  and the cylinder   is oriented counterclockwise as viewed from above.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Use Stokes' Theorem to evaluate  where...

Quiz 13: Vector Calculus

Use Stokes' Theorem to evaluate where and is the triangle with vertices is oriented counterclockwise as viewed from above.

Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives,find where a is the constant vector.

Use Stokes' Theorem to evaluate is the part of the paraboloid that lies inside the cylinder oriented upword.

Set up,but do not evaluate,a double integral for the area of the surface with parametric equations

Use Gauss's Law to find the charge contained in the solid hemisphere ,if the electric field is

Use Stokes' Theorem to evaluate where is the circle . is oriented counterclockwise as viewed from above.

Use Stoke's theorem to evaluate C is the curve of intersection of the plane z = x + 9 and the cylinder

A fluid with density flows with velocity Find the rate of flow upward through the paraboloid

Evaluate the surface integral. S is the part of the plane that lies in the first octant.

Suppose that where g is a function of one variable such that . Evaluate where S is the sphere

Match the equation with one of the graphs below.

,where S consists of the hemisphere and the disk in the -plane.

Find the moment of inertia about the z-axis of a thin funnel in the shape of a cone if its density function is

Evaluate the surface integral for the given vector field F and the oriented surface S.In other words,find the flux of F across S. in the first octant, with orientation toward the origin.

Evaluate the surface integral for the given vector field F and the oriented surface S.In other words,find the flux of F across S.

Use the Divergence Theorem to calculate the surface integral ; that is,calculate the flux of across . S is the surface of the box bounded by the coordinate planes and the planes .

The temperature at the point in a substance with conductivity is Find the rate of heat flow inward across the cylindrical

is the surface of the box enclosed by the planes where are positive numbers.

Evaluate the surface integral.Round your answer to four decimal places. S is surface

Use Stokes' Theorem to evaluate where is the curve of intersection of the plane and the cylinder is oriented counterclockwise as viewed from above.